Jean Lévine

Displacements of a Linear Motor With Oscillating Masses

The aim of this chapter is to show, on a barely undamped oscillating system, roughly approximated by a simple single-input linear model, the importance of feedforward design to simultaneously achieve, with the same smooth controller (no switch), high precision positioning, fast displacements, and robustness versus modelling errors, objectives which are, at first sight, antinomic. We consider a linear motor moving along a rail and, in a first step, related by one flexible rod to an auxiliary mass (see Figure 10.1), and in a second step, by two different flexible rods to two different auxiliary masses (see Figure 10.8). The motor is assumed to be controlled by the force it delivers. All along this study, we assume that the motor position and speed are measured in real time with an arbitrary accuracy, but that the auxiliary mass positions are not measured. Our aim is to generate fast rest-to-rest displacements of the motor with a motor positioning as accurate as possible. Indeed, when the force is non zero, the motor accelerates, producing oscillations of the auxiliary bodies, and, even if their masses are small compared to the motor’s mass, the flexible rods transmit oscillations to the motor, and every controller reaction needed to compensate the corresponding deviation from the motor set point, not only results in a deterioration of the positioning accuracy but in energy expenses and thermal losses. Thus, it is clear that cancelling the oscillations of the auxiliary bodies might be useful. What is less obvious is that, at the same time, the robustness of the controller versus modelling errors is not deteriorated.